Separation of magnetic field lines

The field lines of magnetic fields that depend on three spatial coordinates are shown to have a fundamentally different behavior from those that depend on two coordinates. Unlike two-coordinate cases, a flux tube in a magnetic field that depends on all three spatial coordinates that has a circular cross section at one location along the tube characteristically has a highly distorted cross section at other locations. In an ideal evolution of a magnetic field, the current densities typically increase. Crudely stated, if the current densities increase by a factor sigma, the ratio of the long to the short distance across a cross section of a flux tube characteristically increases by e(2 sigma), and the ratio of the longer distance to the initial radius increases as e(sigma). Electron inertia prevents a plasma from isolating two magnetic field structures on a distance scale shorter than c/omega(pe), which is about 10cm in the solar corona, and reconnection must be triggered if sigma becomes sufficiently large. The radius of the sun, R-circle dot = 7 x 10(10)cm is about e(23) times larger, so when sigma greater than or similar to 23, two lines separated by c/omega(pe) at one location can be separated by the full scale of any magnetic structures in the corona at another. The conditions for achieving a large exponentiation, sigma, are derived, and the importance of exponentiation is discussed. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4765352]